Positronium formation in 2S-state in atomic hydrogen

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1 Indian Journal o Pure & Applied Physics Vol. 46, April 008, pp Positronium ormation in S-state in atomic hydrogen V S Kulhar* Department o Physics, University o Rajasthan, Jaipur Received August 007; accepted 8 February 008 The problem o positronium ormation in its irst excited state (s) in positron-hydrogen collision has been investigated using a ormulation based on three particle scattering. Second order matrix elements have been calculated taking s intermediate state o hydrogen atom into account. The eect o inclusion o second order terms is to raise the total crosssection results over Born results. The cross-section results have been compared with the results o calculations based on irst Born, the two state approximation and convergent close coupling approach in the energy range Ryd and with other available results in an energy range ev. Keywords: Positronium, Atomic hydrogen Introduction The problem o positron-hydrogen scattering is o immense interest to researchers or its being a real three body problem with all its two body interactions known. Unlike the e -H problem, it is also ree rom exchange eects. Recent experiments, on positronium (Ps) ormation have given impetus to these works. The problem o Ps ormation in ground state in e + -H scattering has been exhaustively reviewed both rom theoretical and experimental point o view by Humberston and Griith 4. Ps ormation in ground state takes place or e + having energies above a threshold o 0.5 Ryd. Above threshold, elastic scattering competes with Ps ormation. As the e + energy increases a number o new channels are opened and processes e.g. excitation o hydrogen, capture o e + in higher excited state o Ps and even ionization o hydrogen can take place. In principle, an ideal theory should take into account all available channels and also their coupling. The process o Ps ormation at low and intermediate energies is tackled using close coupling approximation (CCA) methods 5-. Sarkar and Ghosh 5 in their method have retained ground and irst excited state o hydrogen and Ps atom in their expansion o total wave unction. Kernoghan et al. 6 have made use o CCA with states in their expansion. Mitroy 7 has reported a comprehensive calculation which made use o a large L basis o 8 states involving e + - H channels. 8 states basis was supplemented with Ps (s), Ps (s) * vskulhar@hotmail.com and Ps (p) channels. Mitroy 7 has also concluded that CCA with the inclusion o Ps states can describe most o the physics o e + -H systems. Kadyrov and Bray 8 made use o two centre convergent close coupling approach (CCCA).They made use o the method or calculating Ps ormation cross-sections in ground and irst excited state (s, p).yamanaka and Kino 9 used time dependent coupled channel method or Ps ormation in the energy range (6.8-5eV). Kamali and Ratnavelu 0 also obtained Ps ormation cross-sections or low energies using coupled channel optical method. Kar and Mandal used multichannel Schwinger s principle to study Ps ormation in the energy range ev.total Ps ormation crosssections 8- obtained were in close agreement with experimental results o Zhao et al. The experimental results, reported were or total Ps ormation cross-section. These results included contributions o Ps ormation cross-section in ground as well as excited states. A suggestion or measurement o excited states captures cross-sections was made by Kernoghan et al 6. Knowledge o these cross sections is essential or their utility in ine tuning Ps beams which are used in dierent branches o science. Thus, due to non-availability o experimental data or excited states, theoretical data becomes important both as a guide to experimentalist and also or making a meaningul comparison with experimental data on available total cross-sections. In the calculation o e + capture in hydrogen, it has been shown that contribution o e + capture into higher states o Ps to total Ps ormation cross-section,4 alls

2 46 INDIAN J PURE & APPL PHYS, VOL. 46, APRIL 008 o as /n. So major contributions or excited states comes rom Ps ormation in irst excited state (n=) state. Even in n= capture, it was noted that contribution o p capture diminishes aster in comparison with s state with the increase o energy 6,8,5. Hence, study o s capture process is important in e + capture process at high energy. Ps ormation in excited states has been studied by several researchers 5-,4-7. At low and intermediate energies, calculations based on CCA method with the inclusion o direct and rearrangement channel are comprehensive and provide reliable theoretical data 5. At higher energies, a ew ormulations or Ps ormation in excited states exist. It is a FBA calculation 5 with the inclusion o non-orthogonality eect. Saha and Roy 4 used FBA and irst order exchange-approximation or calculation o Ps ormation into arbitrary excited states. At high energies, there is a vide disparity in cross-section results. The high energy behaviour o Ps ormation cross-section is ar rom settled. In ion-atom collisions, it is well known that second order Born term is o vital importance in determining the asymptotic behaviour o capture cross-section. Basu and Ghosh had used a simpliied orm o second Born to study Ps ormation in ground and irst excited state. Tripathi and Sinha 6 made use o Glauber eikonal approximation to study the same problem. Mandal and Mukherjee 7 made use o boundary corrected continuum intermediate state approximation to study Ps ormation in ground and irst excited state. e + -H collision is a three particle scattering problem. One o the approaches or analysis o this system is to make use o Faddeev s 8 ormulation. In this paper, we employ Newton s 9 three particle scattering equations to study Ps ormation in s state. These equations can also be related to Faddeev equations.newton s equations were earlier used by Majumdar et al 0. and us to study Ps ormation in ground state. In this ormulation, one takes transition operator expansion and in this, terms up to second order are taken in or the calculation. Second order term involves Green s unction. It is expanded as a series, the terms in the series are product o plane waves and target (hydrogen) bound states. In our ormulation, we expand Green s unction taking only s intermediate state into account. The intermediate state creates a weak attractive polarization potential or e + and its eect has been studied in this paper. Theory Let incident particle be and (, ) orm a bound system. In our case is positron (e + ) while and are electron (e - ) and proton (p), respectively. We study the ollowing rearrangement (Ps-ormation) process: + (, ) (, ) + () Let T denote the transition operator, then T is an operator corresponding to the elastic process (, ) (, ). T is the operator or Ps ormation process. Let V i denote interaction o j th particle with k th particle and Hi = Vj + Vk. Making use o three particle scattering equations, the transition operators or elastic and Ps-ormation processes are written as: T = H + H G T () + T = H + H G T () + G =, H = H + V + (E H + i ε) 0 (4) In Eq. (), we use T H and obtain T H H G + = + H. We take this as our second order approximation or transition operator or Ps ormation process. Green s unction G + is represented using spectral representation in terms o plane waves and hydrogen wave unctions. The transition matrix elements or Ps ormation process are obtained using ollowing equation: < T i >=< H i >+ < H n k > n,k + < n k G % n k % >< n k % H i > (5) In principle, all intermediate states (discrete and continuous) ought to be taken in the summation. However, or Ps ormation in s state, we are taking only n =, term or simplicity. One may also notice that the term involving n= is smaller than n= since <sk G + < sk is larger than <sk G + sk >or given incident energy. It may be mentioned that in our calculation or Ps ormation in ground state, s and s states o hydrogen atom were taken in the representation or Green s unction.it was ound that at higher e + energies 00 ev, contribution o s term was insigniicant. In atomic units, the initial state o the system is:

3 KULHAR: POSITRONIUM FORMATION IN ATOMIC HYDROGEN 47 i k.r % % r e e < r,r i % % > = (6) π The inal state is: ik %.(r% + r % )/ r r /4 e ( )e < r,r % % > = (7) 6 π Dierential cross-section is obtained using ollowing equation: dσ μμ k ( π a ) = < T i > (8) dω 4π k i 0 i μ i and μ are reduced masses o incoming and outgoing particles.. First order matrix elements In the irst order, one retains only irst term in the expansion in Eq. (). Matrix element obtained corresponds to the irst Born approximation (FBA). π < H i >= [(k% k % ) + ] /8 (k% k ) (k k ) + % + % % 6 6 < H i >= /8 π [(k% k % ) + ] (k% k) % + (k% k) % dx x ( x) EF EF 4EF 5 x E F E F 8E F 6EF Here E = k% k % ( x/) + F and F = x ( x)k /4 + ( 5x/6).. Second order matrix elements The second order matrix element is < H G H i > + = < H sk % >< sk H % i > dk % π + ε ( ) (E Ek i ) (0) () < H sk % > is obtained rom irst order matrix elements by replacing k by k. 4 π 4 < s k % H i >= + () [(k% k) % + 4] [(k% k) % + 4] dq + π k q k + q + (k k + q) + 6 /8 (k% k% + q) + % 6 (9) Angular integration is carried out using Feynman s parameterization technique, and q integral is evaluated using the method o contour integration. Angular part o k % integration is carried out using Feynman s parameterization technique while k integral is evaluated using the method o contour integration. < H G H i > + dy ( x) 5x = ( 4) x dx Q 8 F F ( x) 9x R+ 4( y)r F 64F { } [ + 8( y)r] yr

4 48 INDIAN J PURE & APPL PHYS, VOL. 46, APRIL 008 ( x) ( x) + + F F [ y)r ] y R ( 4 x ( x) y [ R4+ 6( y)r5] 6 F Here R i = I i + i I i Q% = k % ( y) + k% y ( x/) D = k ( y) + 4( y) + y [k ( x/) + F ] E = D - Q () compared with FBA and TSA results. It is seen that in the whole range o energies considered, our results or orward dierential cross-sections and total crosssection are higher in comparison with FBA and TSA results. The peaks in the orward dierential and total cross-section results appear at. and.5 Ryd, respectively. The peaks in total cross-section (s) or both Tripathi et al. s and Mandal et al. s work appeared at an energy less than. Ryd. In Sarkar and Ghosh 5 as well as Kernoghan s 6 work the peak in the total cross-section or s capture appeared at an energy o.57 Ryd. In the works o Kadyrov et al. 8 I ij are deined in appendix. Matrix elements in the second order are obtained by summing contributions o irst and second order terms. Results and Discussion We have computed dierential cross-section at several scattering angles or Ps ormation in s state or an energy o.0 Ryd (see Fig. ). In the representation or Coulomb Green s unction only, s state o hydrogen atom was taken. The results obtained are compared with the FBA and our earlier results based on a two state approximation (TSA) method 5. It is ound that as a result o inclusion o second order term in the matrix element, the dierential cross-section is enhanced in the orward direction. It is.7% higher in comparison with the FBA. However, it is.% lower in comparison with TSA result. Unlike in the case o FBA and TSA results, dip (zero) in the dierential cross-section does not appear in our calculation. Instead a minimum in the dierential cross-section was obtained. Minima in the dierential cross-section were also obtained in the calculations o Tripathi and Sil 6 and in Sarkar and Ghosh s work 5 also. In Mandal et al 7. s calculation a structure (in dierential cross-section) consisting o two minima was seen or high energies. It may be noted that in our calculation or ground state with this method a minimum in the dierential cross-section was observed. This eature also appears in works o Kamali et al 0. and Kar et al s calculations. We have also computed orward dierential and total cross section results or Ps ormation in the sstate in the energy range o Ryd (Figs and ). The cross-section results obtained are also Fig. Dierential cross-sections at k =.0 Ryd as a unction o scattering angle Fig. Forward dierential cross-section as a unction o energy

5 KULHAR: POSITRONIUM FORMATION IN ATOMIC HYDROGEN 49 based on CCC approach, the peak appeared at. Ryd. The peak in the total cross-section (capture into all states) appears in energy interval o.-.5 Ryd. At.0 Ryd, our result or orward dierential cross-sections is.4% higher in comparison with FBA while it is % lower as compared to TSA result. The total cross-section results at.0 Ryd is.% higher over FBA result and is.9% lower in comparison with TSA result. Comparison o total cross-section results with Kadirov et al. s 8 calculation shows a marked dierence. At low energies, our Fig. Total cross-section or Ps ormation (s state) as a unction o energy. FBA: First Born approximation, TSA: Two state approximation, CCCA: Convergent close coupling approach results are signiicantly higher in comparison with Kadirov et al. s results. At energy o Ryd, our results are about three times higher. At 7 Ryd, these are about two times higher. As we move to higher energies, dierence is diminishing (Fig. ). At low energies, our results may not be reliable. At such energies, higher intermediate states e.g. s, p as well as higher order terms in the transition operator [see Eq.()] will also contribute to the Ps ormation process, these terms were not taken into account in the present calculation. By ignoring such terms at low energy, inaccuracy introduced in cross-section calculations will be large. However, at high energy contributions o ignored terms will be small and thus, percentage inaccuracy introduced will be small. The inclusion o higher intermediate states in our ormulation pauses algebraic problems and has not been attempted. It may be seen that the eect o s term is signiicant even at an energy o 0 Ryd. The total cross-section result at 0 Ryd is 4% higher over FBA results and 7.6% higher over TSA result. It may be noted that the s contribution keeps on increasing as energy increases. For instance even at energy o 50 Ryd, our result is 59% higher in comparison with the FBA result. For Ps ormation in ground state contribution o s term diminished with the increase o energy, whereas s contribution was signiicant even at high energies. Due to similarity o the problem o Ps ormation in s and s states, it is anticipated that inclusion o s term in Eq. (5) will lower the cross-section rom present values. However, Table Total cross-sections ( πa ) or Ps ormation in s state. B: irst Born, P: Present result, TSS: Tripathi, 0 Sinha and Sil, M.M.M.: Mandal et al. and BG: Basu and Ghosh. Energy (ev) σ B σ P σ TSS σ MMM σ BG 0.85 (-). (-).0 (-) (-).88 (-).4 (-) (-) 7.55 (-) 6.88 (-) -.65 (-) (-) 4.58 (-).94 (-) (-).88 (-).4 (-) (-).8 (-).44 (-).694(-).5 (-) (-) 8.7 (-) 6.08 (-) 8.69(-).4(-) 5.84 (-).486 (-).9 (-) (-).65 (-).07 (-) (-4) 8.5 (-4) 5.6 (-4) (-4) 4.74 (-4).79 (-4) 5.8(-4) 5.07 (-4) (-5) 7.8 (-5).75 (-5) (-5) (-5).7 (-5) 8.44 (-6) (-6) 5.5 (-6) (-6) 5.4 (-6) (-8).7 (-7) -.(-7) (-9).889 (-9) -.06(-9) -

6 50 INDIAN J PURE & APPL PHYS, VOL. 46, APRIL 008 at high energies the eect o inclusion o s term is likely to be insigniicant. We have made calculations or still higher energies and in Table, we make comparison o our results or total cross-sections with other high energy calculations,6,7. Tripathi et al s. 6 results based on eikonal approximation were lower than FBA results. Our results are higher than Tripathi et al s results or the whole range o energy (0-000 ev) considered. The results o Mandal et al. 7 based on boundary corrected continuum intermediate state (BCCIS) were signiicantly larger than FBA results and are also larger rom our results in the energy region (00eV- kev). The results o Basu and Ghosh are based on a orm o second Born approximation and are higher rom our results or lower energies but become smaller or energies o the order o 00 ev and higher. They are also higher than FBA results in the whole range o energy considerd. CCA calculations are or low and intermediate energies and thus, a comparison o our results could not be made or higher energies with them. It is seen that at high energies, there is wide disparity in results reported or s capture. Merits o dierent calculations can be judged by comparing these results with experimental results. Becker et al. have reported the experimental results or Ps ormation cross-section which are higher in comparison to FBA results. Zhou et al. have also reported the experimental results or Ps ormation in hydrogen and also or hydrogen molecules. Whereas our results are partial (Ps ormation in s-state), they have measured total crosssections including contribution o excited states. Thus, a comparison with their results could not be made. Since present calculation is similar to our earlier calculation, we expect results similar to s capture process, i.e. the contribution o other states s, p -- --will diminish with increase o energy and become insigniicant at high energies and our method will be suitable at such energies. Reerences Zhou S, Li H, Kauppila W E et al., Phys Rev A, 55 (997) 6. Becker D, Lynn K G, Raith W et al., Phys Rev Lett, 68 (99) 690. Humbertson J W, Adv At Mol Phys, (986). 4 Griith T C, Adv At Mol Phys, (986) 7. 5 Sarkar N K & Ghosh A S, J PhysB, 7 (994) Kernoghan A A, Robinson D J R, Mary T McAlindent & Walters H R J, J Phys B, 8 (995) Mitroy Jim, At Mol Opt Phys, 9 (996) L6. 8 Kadyrov Alisher S & Bray Igor, Phys Rev A, 66 (00) Yamanaka Nobuhiro & Kino Yasusi, Phys Rev A, 64 (00) Kamali M Z M & Ratnavelu Kuru, Phys Rev A, 65 (00) Kar S & Mandal P, Phys Rev A,59 (999)9. Basu M & Ghosh A S, J Phys B, (988) 49. Oppenheimer J R, Phys Rev, (98) Saha B C & Roy P K, Phys Rev A, 0 (984) Kulhar V S & Shastry C S, Can J Phys, 56 (978) Tripathi S, Sinha C & Sil N C, Phys Rev A,9 (989) Mandal C R, Mandal M & Mukherjee S C, Phys RevA, 44(99) Faddeev L, Mathematical aspects o the three body problem in Quantum scattering theory (Dave, Newyork), Newton R G, Scattering theory o waves and particles (McGraw Hill, New York), 966, Majumdar C K & Rajagopal A K, Phys Rev, 8 (969)44. Kulhar V S, Pramana, (979).